Now the world-line passing through P will be described by a substantial
point with the constant _mechanical mass m_. Let us call _m-times_ the
velocity-vector at P as the _impulse-vector_, and _m-times_ the
acceleration-vector at P as the _force-vector of motion_, at P.
According to these definitions, the following law tells us how the
motion of a point-mass takes place under any moving force-vector[32]:
_The force-vector of motion is equal to the moving force-vector._
This enunciation comprises four equations for the components in the four
directions, of which the fourth can be deduced from the first three,
because both of the above-mentioned vectors are perpendicular to the
velocity-vector. From the definition of T, we see that the fourth simply
expresses the “Energy-law.” Accordingly _c²_-_times the component of the
impulse-vector in the direction of the t-axis is_ to be defined as _the
kinetic-energy_ of the point-mass. The expression for this is
_mc²_ _dt_/_d_τ = _mc²_ /√(1 - _v²_/_c²_)
_i.e._, if we deduct from this the additive constant _mc²_, we obtain
the expression ½ _mv²_ of Newtonian-mechanics up to magnitudes of _the
order of_ 1/_c²_. Hence it appears that _the energy_ depends _upon the
system of reference_. But since the _t_-axis can be laid in the
direction of any time-like axis, therefore the energy-law comprises, for
any possible system of reference, the whole system of equations of
motion. This fact retains its significance even in the limiting case c =
∞, for the axiomatic construction of Newtonian mechanics, as has already
been pointed out by T. R. Schütz.[33]
From the very beginning, we can establish the ratio between the units of
time and space in such a manner, that the velocity of light becomes
unity. If we now write √-1 _t_ = _l_, in the place of _l_, then the
differential expression
_d_τ² = -(_dx²_ + _dy²_ + _dz²_ + _dl²_),
becomes symmetrical in (_x_, _y_, _r_, _l_); this symmetry then enters
into each law, which does not contradict the _world-postulate_. We can
clothe the “essential nature of this postulate in the mystical, but
mathematically significant formula
3·10⁵ _km_ = √-1 Sec.
V
The advantages arising from the formulation of the world-postulate are
illustrated by nothing so strikingly as by the expressions which tell us
about the reactions exerted by a point-charge moving in any manner
according to the Maxwell-Lorentz theory.
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